Question

The derivative of cos^–1(2x² – 1) w.r.t. cos^–1x is ______.

Options

• 2

• `(-1)/(2sqrt(1 - x^2)`

• `2/x`

• 1 – x² 

Answer 1

The derivative of cos^–1(2x² – 1) w.r.t. cos^–1x is 2.

Explanation:

Let y = cos^–1(2x² – 1) and t = cos^–1x

Differentiating both the functions w.r.t. x

`"dy"/"dx" = "d"/"dx" cos^-1 (2x^2 - 1)` and `"dt"/"dx" = "d"/"dx" cos^-1x`

⇒ `"dy"/"dx" = (-1)/sqrt(1 - (2x^2 - 1)^2) * "d"/"dx" (2x^2 - 1)` and `"dt"/"dx" = (-1)/sqrt(1 - x^2)`

= `(-1.4x)/sqrt(1 - (4x^4 + 1 - 4x^2)` and `"dt"/"dx" = (-1)/sqrt(1 - x^2)`

= `(-4x)/sqrt(1 - 4x^4 - 1 + 4x^2)`

= `(-4x)/sqrt(4x^2 - 4x^4)`

= `(-4x)/(2xsqrt(1 - x^2)`

⇒ `"dy"/"dx" = (-2)/sqrt(1 - x^2)`

Now `"dy"/"dx" = ("dy"/"dx")/("dt"/"dx")`

= `((-2)/sqrt(1 - x^2))/((-1)/sqrt(1 - x^2))`

= 2.